Saturday, 16 December 2017

SLOPE OR GRADIENT 1


Find the slope of a line 7x – 8y + 10 =0.

Solution

7x – 8y + 10 =0

We strive to re-write it in the form y=mx+c.

– 8y + 10 = -7x    [shifting 7x to the right side.]

– 8y = -7x – 10    [shifting 10 to the right side.]

– 8y = -7x10    [dividing by -8 all over]
 -8      -8      -8

y = 7x/8 + 5/4 which is like y=mx + c

Hence Slope is 7/8 


TRY THIS……………………….



Find the slope of a line 9x – 4y + 14 =0.

LOGARITHMS 2


Evaluate Log100 -  log 0.001 +  log 0.0001

Solution

= Log100 -  log 0.001 + log 0.0001

= Log102 -  log 10-3 + log 10-4

= 2Log10 - (-3 log 10) + (-4log 10)

= (2x1) - (-3x1) + (-4x1)

=2 - (-3) –4

= 2 + 3 - 4

=1
  
Hence Log100 -  log 0.001+  log 0.0001 = 1


TRY THIS……………………… 



Evaluate Log100 -  log 0.000001 +  log 0.00000001

GEOMETRY 1


The interior of a regular polygon is 1700. Find the number of sides

Solution

i = (n-2)1800
          n

But  i=1700.

1700 = (n-2)1800
                 n

170 x n = (n-2)1800 x n  multiplying by n both sides.
                  n

170n = (n-2)1800   

170n = 180n-3600

3600  = 180n- 170n

3600  = 10n

n = 36 after dividing by 10 both sides.    

Hence number of sides = 36            


TRY THIS……………………….



The interior of a regular polygon is 1500. Find the number of sides



EXPONENTIALS 1



If 810 = 32a-4; find a

Solution

810 = 32a-4 

(23)10 = (25)a-4 

230 = 25(a-4) 

230 = 25a-20
230 = 25a-20  [Same bases both sides cancels out]
30 = 5a – 20
30 + 20 = 5a
50 = 5a
50 = 5a
5      5

10 = a

Hence a=10

TRY THIS……………………….


If 520 = 625w-8; find w

PERFECT SQUARES 1



What must be added to x+ 14x to make the expression a perfect square?

Solution

a=1, b=14,c=?

b2 = 4ac

(14)2 = 4 x 1 x c

196 = 4c
196 = 4c
 4       4

49 = c

Hence number to be added is 49

TRY THIS……………………


What must be added to x+ 32x to make the expression a perfect square?   

Sunday, 10 December 2017

LOGARITHMS 1


If Logax = 4.9, find Loga(1/x2)

Solution

= Loga(1/x2)

= Logax-2

= -2 x Logax

= -2 x 4.9

= -9.8


TRY THIS………..


If Logaw= 8.6, find Loga(1/w2)




QUADRATICS 2


Find the maximum value of the quadratic equation:3-2t-10t2.

Solution

a=-10, b=-2, c=3

Maximum = 4ac-b2
                        4a

Maximum = (4 x -10 x 3) - (-2)2
                          4(-10)

Maximum = (-120)- 4
                         -40

Maximum=  -124
                      40

Maximum = -31
                     10
Hence maximum value is -31/10


TRY THIS………………………………..



Find the maximum value of the quadratic equation:7+28t-6t2.



Saturday, 9 December 2017

QUADRATICS 1


Given that one of the roots of the equation 5x2 + m(x+1) + 5 = 0 is 7, find m.

Solution

Substitute x=7, in the above equation.

5(7)2 + m(7+1) + 5 = 0
(5x49) + (m x 8) + 5 = 0
245 + 8m + 5 = 0
8m + 5 = -245
8m = -245 - 5
8m = -250
8m = -250
8         8
m=-125/8

Hence m=-125/8

TRY THIS…………………………..


Given that one of the roots of the equation 3x2 + e(x-2) + 3 = 0 is 4, find e.

POLYGONS 1



A regular polygon has 85 sides. Find the total angles of that polygon.

Solution

n = 85

Total angles = (n - 2)1800

                    = (85 – 2)1800

                    = 83 x 1800

                    = 149400

Total angles = 149400


TRY THIS……………………….


A regular polygon has 48 sides. Find the total angles of that polygon.




Monday, 20 November 2017

STATISTICS 1



In Biology test the following marks were recorded.

marks
10-19
20-29
30-39
40-49
50-59
No. of students
3
4
9
6
3


Calculate the mean.

Solution

In this question we are going to apply the method of assumed mean.

Here you are required to produce the frequency distribution table.

Let us take 44.5 as our assumed mean (A). Take away the assumed mean from each class mark.


Class interval
Class mark
(x)
d=x-A
Frequency
(f)
fx
fd
10-19
14.5
-30
3
43.5
-90
20-29
24.5
-20
4
98
-80
30-39
34.5
-10
9
310.5
-90
40-49
44.5
0
6
267
0
50-59
54.5
10
3
163.5
30

∑f = 25
∑fx = 882.5
∑fd = -230


Mean = A +  ∑fd
                      ∑f

Mean = 44.5 + (-230)
                            25

Mean = 44.5 -  9.2
                           
         
Hence Mean =  35.3


TRY THIS………….

In Civics test the following marks were recorded.

marks
10-14
15-19
20-24
25-29
30-34
No. of students
  6
  10
 14
12
  8



SIMPLE INTEREST 1


Rugaya deposited the amount of 90,000/= in a bank for 5 years and got a profit of 22,500 /=. Find the interest rate?

Solution

I = 22,500 /=, P = 90,000/=, T = 5 years, R = ?

I  = PRT
      100

22,500  = 90,000 x R x 5
                      100

22,500  = 90,000 x R x 5
                      100

22,500  = 900 x R x 5
                      
22,500  = 4500R
           
522,500  = 4500R
   4500       4500


Hence the interest rate was  5%

TRY THIS………….


Rweyendera deposited the amount of 12,000/= in a bank for 5 years and got a profit of 2100/=. Find the interest rate?


PROBABILITY 1


A bag contains 12 red balls and 8 blue balls. Two balls are taken from the bag. What is the probability that they are both red?

Solution

(This is a problem with replacement)

n(R) = 12, n(B) = 8, n(S) = 20

P(R) = n(R)
            n(S)

1st pick = 12/20
2nd pick = 12/20 as well.


P(RR) =  12      x    12
               20            20

P(RR) =         
                25          

Therefore Probability of drawing a red ball is 16/81     


TRY THIS ..............


A bag contains 8 purple stones and 14 blue stones. Two stones are taken from the bag. What is the probability that they are both blue?



BODMAS 1


Evaluate 74 x (57 – 37) ÷ (845-843)

Solution

We apply BODMAS.

= 74 x (57 – 37) ÷ (845-843)

= 74 x 20 ÷ (845-843) [After dealing with 1st brackets]

= 74 x 20 ÷ 2     [After dealing with 2nd brackets]

= 74 x  10     [After dividing]

= 740     [After multiplying]

Hence  74 x (57 – 37) ÷ (845-843) = 740

TRY THIS……………….


Evaluate 60 x (93 – 13) ÷ (89-49)


LOGARITHMS-3


If log6(2x + 17)=2; find x.

   Solution

Log9(2x + 17)=2

(2x + 17)=62           
                                            
2x + 17=36

2x =36 – 17

2x = 19

2x =   19   
2         2

x = 9.5

Hence x = 9.5

TRY THIS………………


If log11(2u - 7)=2; find u


Sunday, 19 November 2017

INEQUALITIES 1






PERCENTAGES 1


A man got a profit of 7300/= after selling an item. Find the buying price if the percentage profit was 10%.

Solution

%’ge profit = Profit   X  100    where B. P. represents Buying Price.
                         B.P

10% = 7300   X  100   
            B.P.

B.P. x 10% = 730,000   x B.P.  [multiplying by B.P. both sides]
                       B.P.


B.P. x 10 = 730,000   

        
B.P. x 101 =   730,000     [dividing by 10 both sides]  
  110                   10

B.P. =73,000
                      
Hence Buying Price was 73,000/=


TRY THIS………………

A man got a profit of 16,200/= after selling an item. Find the buying price if the percentage profit was 10%.

EXPAND 2


Expand 8w(5w + 4 - w)


Solution


= 8w(5w + 4 - w)

= (8w x 5w) + (8w x 4) - (8w x w)

= 40w2 + 32w - 8w2

= [40w2 - 8w2] + 32w collecting like terms

= 32w2 + 32w answer


TRY THIS………..



Expand 4a(5a + 14 - a)


EXPONENTIALS 1


If 52w (400w) = 1000 ; Find w.

Solution

52w (400w) = 1000

(52)w (400w) = 1000

(25)w (400w) = 1000

(25 x 400)w = 1000

(10000)w = 1000

(104)w = 103   [since 103 = 1000]

104w = 103   (Bases are alike, so they cancel out)

4w = 3

4w = 3
4       4

w = 3/4

TRY THIS…………………………….


NECTA 2004 QN. 4b


If 32t (4t) = 6 ; Find t.